A grocery sack can withstand a maximum of 230 N before it rips. Will a bag holding 15 kg of groceries that is lifted from the checkout counter at an acceleration of 7.0 m/s² hold?
Understand the Problem
The question is asking whether a grocery sack can withstand the forces acting on it when lifting a bag of groceries with a given mass and acceleration. We will approach this by calculating the force using Newton's second law and comparing it to the maximum force the sack can withstand.
Answer
The bag will rip since the total force required is $252 \, \text{N}$, which exceeds the maximum of $230 \, \text{N}$.
Answer for screen readers
The calculated force is $F_{\text{total}} = 252 , \text{N}$. Since $252 , \text{N} > 230 , \text{N}$, the bag will rip.
Steps to Solve
- Calculate the weight of the groceries
The weight ($W$) can be calculated using the formula: $$ W = m \cdot g $$ where ( m = 15 , \text{kg} ) (mass of the groceries) and ( g \approx 9.8 , \text{m/s}^2 ) (acceleration due to gravity).
- Calculate the net force when accelerating
The total force ($F_{\text{total}}$) required to lift the groceries while accelerating can be calculated using Newton's second law: $$ F_{\text{total}} = m \cdot (g + a) $$ where ( a = 7.0 , \text{m/s}^2 ) (acceleration).
- Combine the calculations
Substituting the known values into the force equation: $$ F_{\text{total}} = 15 \cdot (9.8 + 7.0) $$
- Compute the resultant force
Calculate the total force:
- First, add ( g ) and ( a ): $$ 9.8 + 7.0 = 16.8 , \text{m/s}^2 $$
- Now multiply by the mass: $$ F_{\text{total}} = 15 \cdot 16.8 $$
- Determine if the sack can hold the force
Finally, compare the calculated force to the maximum force the grocery sack can withstand:
- If ( F_{\text{total}} > 230 , \text{N} ), the sack will rip. If ( F_{\text{total}} \leq 230 , \text{N} ), the sack will hold.
The calculated force is $F_{\text{total}} = 252 , \text{N}$. Since $252 , \text{N} > 230 , \text{N}$, the bag will rip.
More Information
In this scenario, the grocery sack is not able to withstand the force when lifting the groceries with the given acceleration. Understanding the forces acting on objects during acceleration is crucial in many applications, including engineering and safety design.
Tips
- Forgetting to include the effect of gravity when calculating the total force.
- Confusing mass with weight; remember weight is the force due to gravity.
- Not converting units properly if dealing with both metric and imperial measurements.
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